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Research, consulting, and advanced engineering in Ihe fields of NAVAL and INDUSTRIAL HYDRODYNAMICS. Offices and Laboraforv in the Washington, D. C, area: 200 Monroe Street, Rockville, Md.

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TECHNICAL REPORT 117-1

ANALYSIS OP THE BULBOUS BOW

ON SIMPLE SHIPS

by

B. Yim August 1962

Prepared under

Office of Naval Research Department of the Navy Contract Nonr-33^9(00)

NR 062-266

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TABLE OF CONTENTS

Page No.

LIST OP SYMBOLS 1

LIST OF FIGURES ill

1. INTRODUCTION 1

2. FORMULATION OF THE BOUNDARY VALUE PROBLEM (CASE l) 3

3. FORCE AT DOUBLET POINT 5

4r NONDIMENSIONAL FORM 7

5. EVALUATION OF INTEGRAL R 9 o 6. FORCE AT SOURCE POINT 12

7» TOTAL FORCE AND OPTIMUM PARAMETERS 13

8, REMARKS 15

9- WAVE HEIGHTS (CASE II) 16

IG. HAVELOCK'S FORMULA AND WAVE RESISTANCE 18

11. NONDIMENSIONAL FORM 20

12. OPTIMUM PARAMETERS 21

13. SHIP SHAPES 22

14. DISCUSSION 2h

15. WAVE RESISTANCE OF THE SYSTEM OF A DOUBLET LINE AND A SOURCE LINE (CASE III) 25

16. NONDIMENSIONAL FORM 27

17- OPTIMUM PARAMETERS 28

18. EFFECT OF STERN 30

19. DISCUSSION 32

20. APPLICATION 33

APPENDIX 36

REFERENCES 39

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LIST OF SYMBOLS

-a x coordinate of doublet or doublet line

a. polynomial coefficients of source distribution

b radius of a bulb (approximated as a sphere)

d length of source line

P-, F,, FT Froude numbers with respect to f, d., and L re- I Q L j. n spectively

f depth of a point source below the free surface

f / depths of end points of source line

g acceleration of gravity

G (k )jl Integrals defined by Equations [16] and [14] respectively

K ,K Modified Bessel Functions of the 2nd kind o i

ko = g/V2

L distance between two vertical source lines fore and aft

m total strength of a point source or a source line

m = m/d

r radius of half body

R total wave resistance

R > R , R, ] o s b

RJ x-^ R (i • J ^ n ' int ZX is I Ra (a.b^), ^(a^b^c, ) j

Re = real part of

Parts of Wave resistance defined in Sec- tions 7 and lO.


-11-

Rl [ak jk jn] an integral defined by Equation [12]

V Uniform free stream velocity at oo

yi,Y,z Rectangular right handed coordinate system with origin at free surface., z positive upward^ and x in the direction of the uniform flow velocity V

<t) total velocity potential

^x'^z'^xo Potentials defined in Section 2

|i strength of doublet

fx .|i linear coefficients of strength of doublet in \i=\ix~z\iz

v = v1/V

v^ fictitious frictional force

p mass density of water

CD = (x + a) cos a + y sin a

t, wave height

^ j L wave height due to sources and doublets respectively. S D


-111-

LIST OF FIGURES

FIGURE NO.

1. COORDINATE

2. CONTOUR OP INTEGRATION

3. WAVE RESISTANCE OP HALF BODY WITH OPTIMIZED BULB

4. OPTIMUM DISTANCE BETWEEN SOURCE AND DOUBLET, OP- TIMUM RADIUS OF BULB

5. WAVE RESISTANCE OF A SOURCE LINE AND OPTIMUM DOUBLET LINE AT EACH FROUDE NUMBER

6. OPTIMUM DISTANCE BETWEEN DOUBLET AND SOURCE LINE

7. OPTIMUM RADIUS OF BULB (A POINT DOUBLET)

8. OPTIMUM•DEPTH OF DOUBLET

9. BODY STREAMLINE SHAPE Dim TO A SOURCE LINE AND A POINT DOUBLET

10. BODY STREAMLINE SHAPE DUE TO A SOURCE LINE AND A POINT DOUBLET

11. BODY STREAMLINE SHAPE OF SOURCE AND SINK LINES AND A POINT DOUBLET

12. OPTIMUM DISTANCE a, BETWEEN DOUBLET LINE AND SOURCE LINE

13- OPTIMUM STRENGTH OF DOUBLET LINE

14. WAVE RESISTANCE OF SOURCE LINE AND DOUBLET LINE (BULB)

15. WAVE RESISTANCE FOR SOURCE LINE SHIP INCLUDING STERN

16. WAVE RESISTANCE INTERFERENCE WITH STERN

17. BODY STREAMLINE SHAPE DUE TO A DOUBLET LINE AND A SOURCE LINE

HYDRONAüTICSj Incorporated

ANALYSIS OF THE BULBOUS BOW ON SIMPLE SHIPS

1. INTRODUCTION

The effect of bulbous bows on the wave resistance of

ships was first Investigated by the systematic experiments of

D. W. Taylor (1911 and 1943), E. M. Bragg (1930), and E, P.

Eggart (1935)• It had been generally understood that the de-

crease of resistance due to a bulbous bow is a wave-making

phenomenon. J. G. Thews in 1930-1932 conducted experiments to

determine the conditions for the cancellation of the bow wave

by the bulb. (See H. E. Saunders 1957). In 1928, Havelock

calculated the wave form due to a doublet immersed in a uni-

form stream, and found that a wave trough was formed just aft

of the doublet. Since a deeply immersed sphere is equivalent

to a doublet, W. C, S. Wigley (1936) Investigated this effect

with a mathematical model of a Michell's ship plus a doublet.

By using Havelock's resistance formula (1934b), he demonstrated

the fact that the doublet wave cancels the bow wave and thus

lessens the wave resistance. His analytic work was also sup-

plemented and confirmed by his model experiments. G. Weinblum

(1935) dealt with this problem by expressing the form of a

ship with a bulbous bow in terms of a polynomial according to

Michell's thin ship approximation.

Recently Inui, Takahel and Kumano (i960) observed wave

profiles and found that the wave due to a ship with a submerged

sphere faired into the bow was exactly the superpositio . of the

wave due to the corresponding point doublet and that due to the

hull. They explained the effect of the bulb on the wave resis-

tance of a ship by using the idea of Havelock 's elementary

surface wave (1934a).


The mathematical expression of the wave resistance of

a conventional ship Is extremely complicated. Except for the

special cases of very low or very high Froude numbers It

Is necessary to Integrate a highly oscillating function numeri-

cally. And It Is extremely difficult to Investigate the effect

of varying parameters In the wave resistance equation, There-

fore, only simple models of ships are considered In the present

paper. The advantages of this procedure are: 1) We can analyze

not only the wave resistance but also the effect of the size

and the location of the bulb. 2) We need not be concerned

about the effect of linearizing the boundary condition on the

ship surface. 3) We can Investigate the fundamental relation-

ship between the source and doublet under the free surface.

At first, the simplest system, a point doublet and a

point source. Is considered. The so-called Interference term

of the wave resistance Is calculated by a series expansion.

The optimum distance between two singularities, and the optimum

size and depth of the bulb are obtained. By this procedure we

can show that a remarkable reduction In the wave resistance can

be realized by the use of bulb.

Second, a system consisting of point doublet and a finite

source line Is considered. This Is treated In similar manner to

the first case .

Third, a system consisting of a vertical doublet line

under the free surface and a vertical source line from the free

surface Is considered. The strength of the doublet line Is con-

sidered to vary linearly. However, the optimum distribution of

the strength of the doublet line Is found to be almost uniform.

The Influence of a stern Is also considered using the method of

stationary phase. In this case, the calculation is performed on

F^FONAUTICS, Incorporated

-3-

the I.B.M. l620 Digital Computer.

To obtain the wave resistance, Lagally's theorem Is used

In the first case while Havelock's formula Is used In the second

and third cases. When Lagally's theorem Is used we can see very

clearly how a thrust force Is applied at the point of the doublet,

when It Is located at the proper position In front of the source.

The actual shapes of the low resistance systems are found

by computing the actual stream lines. The bulb Is found to be

very large so that the bulb Itself may be considered to be a bow

rather than an appendage .

This method of analysis Is shown to be applicable to the

general case In which the waterllne of a ship Is expressed as a

polynomial.

Case 1

2. FORMULATION OF THE BOUNDARY VALUE PROBLEM

The origin 0 Is placed on the mean free surface. The x

direction Is the same as that of the velocity of the uniform

stream. The z axis Is directed upward from the surface. The

coordinate axes 0-xyz shown In Figure 1 form a right handed sys-

tem. Symbols are shown in the figures or in the symbol table If

not mentioned In the text. The water Is, as usual, considered to

be incompressible, homogeneous, and inviscld. The motion is

considered to be steady.

A system consisting of a doublet at point (-a,0,-f), a

source at point (0,0,-f) and a sink at point (L,0,-f) in a uni-

form stream will represent approximately the combination of a

bulb and a Rankine ovoid. The wave resistance due to this system

can be obtained by using Lagally's theorem. To investigate mainly

the relation between the bulb and bow, we only need consider the


■4-

system of the doublet at point (-a,0,-f) and the source at point

(0,0,-f). We denote * and <t> as the perturbation potential

due to the point doublet and the point source respectively. The

symbols m and -\i denote the strength of the source and the doublet

respectively.

Denoting R{&,h,c) as the x component of the force at the

point (a,b,c) we obtain by Lagally's theorem (Lagally 1922 or

see Milne Thomson 1956).

R (-a,0,-f) = - 4 TT p M. Ox

a* 10

äx ox (-a^-f)

R (0,0,-f) 2

4 TT p m S*

20

ox ox (0,0,-f)

[1]

[2]

The subscript 0 to ^ means that we exclude the effect of the i

point Itself.

Now we have only find the perturbed velocities at two

points (-a,0,-f) and (0,0,-f). The perturbation potential *

must satisfy Laplace's equation.

V2 * 0

and the boundary conditions on the free surface and at Infinity

as well as near the singularities. The linearized kinematic

boundary condition on the free surface [see Lamb 1945] Is given

by

* = -V C on z=0 z ^x

where £ Is the free surface elevation. If the idea of fict-

itious frlctlonal force [see Lamb 1945 or Lunde 1952] is used.

[3;

[V


-5-

the pressure condition on the free surface Is

* V-g C + v <D = 0 on z = 0 [5]

v being the coefficient of the fictitious frlctlonal force, i

Combining above the two conditions we obtain

4)+k<t)+vtt)=0 [6] xx o z x

on z = 0

where k s g/V2» v - viA-

When the distance from the singularities goes to Infinity,

V * = 0 [?]

In addition, we note the empirical fact that the disturbance

In front of ship decreases very rapidly when -x becomes large.

3. FORCE AT DOUBLET POINT

The solution of our problem [3] with boundary conditions

[6] and [7] can be found In many papers (e.g. Lunde I952) . Us-

ing the Integral representation

-k( |z-f |-1üU) 27r

m m

V(x+a)2 + y2 + (z-f)2 27r 0 0

where CD = (x+a) cos 0 + y sin 6, we have for the potential,

due to only the Images of the doublet, satisfying the conditions

[3], [6] and [7]


M.(x+a)

10 r i 3/'2

J(x+a)2 + y2 + (z-f)2'

k T" oo k(lü5- Z-f ) „ o r r i k sec 9 e x I ''dkd0

-IT 0 k-k sec' o

iv sec

where Re means "real part of". Hence

a* 10

ÖX (x+a)2 + y2 + (z-f):

3/2

3 (x+a):

(x+a)2 + y2 + (z-f): 5/2

k TT

+ Re — / / TT

k2 e k(ia)-|z-fj) dkde

TT 0 k-k sec' o

iv sec

The x component of the perturbed velocity due to the source,

which satisfies Equations [3], [6] and [7] is

/ a* öx / x,0,-f

= m - +

x2+ h f2 3/2

k m TT «> k(ix cos 0 - 2f) ,, ,. 0 r r Ik sec 0 e x ' dkd0 r0 -Re / / [8. TT

-TT 0 k-k sec 0 - iv sec 0


Hence from Equation [l]

R (-a,0,-f) = - h v p \i _, o r r 1 k cos 9 e dkd©

-v 0 k-k sec' o

iv sec 9

+ m 3ac

a2 + 4f2 3/2

a2+4f2 5/2

km v oo , 2 -k(ia cos 0 + 2f) ,, ,_ „ o . r k e v 'dkdö

+ Re - / / IT

-TT 0 k-k sec' o

iv sec [9]

The Integral with respect to k can be evaluated using the contour

Integral as In Equation [11]. After the Integration, v Is set

equal to zero. Then the first Integral can be represented by

means of modified Bessel's functions (see Lunde 1952) as in

Equation [15] I.e.

TT co o -2k f „ /• r 1 k cos ö e dkdö Re / /

-k f , 3 0

TT k J e -TT 0 k-k sec'

0 Iv sec 9

>: s Ko(kof) + 1 + 2rF \ (y^ 4. NONDIMENSIONAL FORM

We may use the relation between p. and the radius of the

sphere b In a uniform stream at infinite depth

. = ^


Similarly

V 2 m = ^ r

where r Is the maximum radius of a half body produced by a

source In a uniform stream.

Then In the nondlmenslonal form as

R R = i 1 I P V2 r2

b = — , P.p = —p= = Froude number, etc f f "V

- a - a = f ' ko k f =

v2 P: r~j,

and dropping bars for convenience we have

R (-a,0,-1) = - bs 3a'

a2 + 4 3/2

a2 + 4 5/2

+ 2bt 8 2

■^ K (k ) + o o

1 + 2 k Ki (k0)f

- Re 2b-;

TT E R

2 0 [10]

where K and K represent the zero order and first order 0 1

modified Bessel functions of the 2nd kind respectively, and


TT M ,2 -kfla cos 0 + 2f) ,. ,_ R = Re / / ^-^— -^- 0 2

-TT 0 k-k sec 0 - Iv sec 0 o

[11]

5. EVALUATION OF INTEGRAL R

Taking a contour as shown In Figure 2 for the Integra-

tion of (l) with respect to k and, noting that v>0 for the resi-

due at the singularity k = k sec2 0 + v sec 0 and later let-

ting v-»0. We obtain

Tr/a

R = Sir o

-2k f sec- 0 k 2 sec4 9 e sin (ak sec 0)d0 o x o

-v/z

+ / / 0 -IT

-211 f TT (k sec2 0 + It) t2 e slnh (at cos 0)d0 Idt

t2 + k 2 sec4

The second Integral of the right hand side can be shown to vanish

because of the odd function, slnh.

Hence

F4R f o 2TT

TT/S -2 k sec" 9 4 - ' o sec 0 sin (ak sec 9) d0 v 0

= Rl ak ,k ,4 o o

[12]


-10-

If we expand sin (ak , sec 9) Into a series of ak sec 6,

2n + 1

Rl[ak .k A] = J e o o J

0

TTA -2k sec2 9 sr^ n (ak ) (-1) (2n + 1)

n=0

x sec2n + 5 0 d 0 [13]

To evaluate the integral

TT/P -2k sec2 9 2v + 1 I = / e 0 sec 0 d 0 V 0

[14]

we put sec 0 cosh — and obtain

-k

i„ -1 / oo -k cosh u

o e

1 + cosh u 2 /

d u = —-—r- G (k ) ^v + 1 vv o

oo -k cosh u , G (k ) = / e 0 (1 + cosh u) du = K (k ) - K (k ) i o i. oooo [15]

where K '(k ) = d K (k )

o o oo d k

K (k ). In addition we can i o

prove easily

oo -k cosh u n G(k)=/e0 (1+ cosh u) du

n 0 Ö

= G , (k ) - G' (k ) n-1 o n-1 o

[16]


Hence we get

-k

K (k ) 1 ov o "v 2V + 1

where

-11-

K' (k ) o v o '

(n)

(K (k ) - K' (k 0 0 o ' 0

(n)

in\ dK (k ;

K (k ) -111 ^^ o v o dh

m d2 K (k ) + 12) 0—^- + (-1)

dh^

dnK (k ) n o o

~ dhn [17]

We now perform the integration of [13] term by term and obtain

2n + 1

Ri(ak .k ,4) = (-1) (ak )' n v o (2n + 1) ! n + 2

n=0

n=0

.l)n(ak )2n+1

(^+1)'. 2 I J(n+2) [18]

This series is proved to be convergent for any ak and k >0 oo

(see appendix). When ak is sufficiently small this series

converges rapidly. In computations^ the following recurrence

relations are of great value.

Gn(h) = 2 + n - 1

h G , (h) - (2n " 3^ G 0 (h) n-1 h n-2 v [19]


-12-

for all integers n. This can be easily proved by Integrating

Equation [15] by parts. In addition

GJh) = K (h)

G (h) = K (h) + K (h; i o i

[20]

On Inserting the value of R obtained by the foregoing pro-

cedure into Equation [10] we finally obtain the non dimen-

sional, horizontal component of the force at the doublet.

6. FORCE AT SOURCE POINT

Similarly^ the force at point (0,0,-f) is obtained

from [2] In dimensional form

i

ox 0,0,-f - M- x+a

(x+a):

(x+a)2 + (z-f)2} A 3/2

0 o o , r Ik sec 0 e v dkd© IT ÖX , ■ ' , -IT 0 k-k sec 9 - iv sec

o 0,0-f

la

3a'

(aMf2)3/2 (a2+4f2)5/2

k u. IT «> , a k(ia cos Ö-Sf),, ,. Or f k e v dkdö + Re -jp / / -ir 0 k-k sec2 9 - iv sec 9

o


■13 ■

20 D o f f Ik sec 0 e dkd0 -T- 1 = Re f ÖX '0,0,-f TT •' /, , , 2 n ■ ' -v 0 k-k sec 9 -iv sec

o

=k m ^K (k f) + K (k f), 0 10 0 1 o

R (0,0,-f) = ^ TT p ml^-^ + 20 'öx ' ox / 0,0,-f

The integrals are evaluated exactly In the same way as In

Sections 3-5-

7. TOTAL FORCE AND OPTIMUM PARAMETERS

The total force, or the total wave resistance Is, In

nondlmenslonal form,

R = R (-a,0,-1) + R (0,0,-1)

= e 2bt

K (k ) + a-n ~ i o o r F,

1 + 2k K (k ) i o

2F K (k ) + K (k )

4 i ov 0 i' 0 ' f

8b3 Ri ak ,k ,4

o o - [21:

where Rl(ak ,k ,4) is given by Equation [18]. We may write o o

alternatively

R = R + R + R, s b int


.14-

where the nondlmenslonal wave resistance (or wave resistance

coefficient) due to a source

-ko r "1

and due to a doublet

R = 2e -k ,6

o b

P. K (k ) + 1 0V 0

1 + 2k 0

and the interference term

8 R Int b3 Rl ak ,k , 4

o o

Hence R, + R, r is the total Influence of the bulb. To obtain b Int

the optimum value of the distance a which makes R minimum we

notice that Rifak ,k ,4) is the only function of a appearing in oo

the right hand side of the above Expression [21]. The optimum

value of a is readily obtained from Equation [18] with the aid

of a numberical - graphical procedure.

The optimum value of b was obtained by differentiating R

with respect to b., putting it equal to zero, and solving for b.

2P 2 Rl ak ,k ,4 o o

r? K (k ) + o o

[22]

1 + 2k K{ko)

I


-15-

The optimum values of a and br / thus obtained are plot-

ted against F^ In Figure 4.

8. REMARKS

With these optimum values of a and b,, R/r2 Is plotted

In Figure 3 versus F2. Here we can see that a considerable

reduction In the wave resistance Is obtained by using an opti-

mum bulb. The force at the point of the doublet In the pre-

sence of the source with optimum values of a and b Is also

plotted In Figure 3• We can see there that the favorable effect

of the bulb Is due to the pulling force (negative) at the

doublet. The force at the point source In the presence of the

doublet Is not shown there, out oovlously It Is a positive

force., which Is the sum of the absolute value of force at the

doublet point and the net force which are both shown In Figure

3. This reveals that the reduction of force on the half body -

bulu combination Is entirely achieved by the creation of a

large bulb thrust and not by the reduction of resistance on

the half-body Itself. In fact, the wave resistance of the

half-body by Itself would appear to Increase In the presence

of the bulb.

The wave resistance of a source has a singularity at

f=0. The wave resistance coefficient C Increases as f/r de- w '

creases by the factor (f/r) at a fixed Froude number as shown

In Figure 3 or Equation [10]. Although the shape of the body

formed by a source near the surface looks more like a ship

shape, this singularity prevents us from approaching It In this

way.

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Prom the radius of the bulb b shown in Figure 4 we

obtain the strength of the doublet [x = V b3/2. This relation

is used only for approximating the size of bulb. The doublet

here Dy no means results in a sphere. There is not only the

Influence of the free surface but also that of the source to

alter its shape from that of the sphere. The form of the

body will be discussed, in Section 14 later. Although the dis-

tance, a, between the doublet and the source seems to be large

compared to b it is highly probable that the resulting bulb and

bow are connected by faired lines (See Figure 9A)•

CASE II

9- WAVE HEIGHTS

We now consider the same problem as in Case I except that

the point source Is replaced by a vertical line source extend-

ing from the free surface to depth d. In order to find the

wave resistance it Is now more convenient to use Havelock's

formula, (Havelock, 193^) rather than Logally's theorem. With

the former method it Is necessary to determine the wave height

at a large distance down-stream of the body.

As shown In Equation [8] the x component of perturbed

velocity of the flow due to a point source at depth f with 1

strength m is for large x

>., mk TT 00

ox' x,y,0 TT ^ J0

k(ia)1 - fj 1 k sec 9 e dkd0

k-k sec2 0 - iv sec 9 0

[23]

r HYDRONAUTICS, Incorporated

-17-

where CD =xcos0 + ysin0 i

Hence, if we consider a vertical source line with uniform

strength m per unit length, (or m=m d), then on the free i i

surface we have for large x

a* Ox

ra k ^(i^-fj

d TT Co ik sec 9 e dkdö df

x.y^O u Re / / /

0 -IT 0 k-k sec' o

iv sec 0

mk

IT Re / /

-kd kiüo1 (e -1) i sec 0 e dkd9

-IT 0 k-k sec" 9 - iv sec o

ir/a -k d sec2 0 m k / (l - e ) sec 0 cos (k x sec 9) i o 0 o

X cos (k y sin 0 sec" 0) d9 [24]

using contour integration as was done in [ll] and [12]. Hence

the wave height far from the source line is, from [5] with

v = 0 1

m v/2 -k d sec"0 C = 8 -TT J (1 - e 0 ) sec 9 cos (k x sec 0) ^s V -L v v o '

x cos (k y sin 0 sec2 0) d0 v 0 [25]

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• 18-

Slmllarly^ the wave height due to a doublet at (-a,0,-f) Is,

for large x

8k 2H V: ^- V

-k fsec-9 2 o

/ sec 9 e 0

[sln(k x secö) cos (k a sec0) v o J K o '

+ cos (k x secö) sin (k a sec9)] cos(k y sin 9 sec;a9)d9 [26^

10. HAVELOCK'S FORMULA AND WAVE RESISTANCE

Suppose the wave height at large x.,

ir/a £, = J (P sin A cos B + P cos A sin B + P cos A cos B

0 1 2 3

+ P sin A sin B)d9 4

[ST!

where A = k x sec 9. B = k y sin 9 sec 9. PJ is a function 00 i

of 9 and other physical parameters. Then Havelock's formula

(193^) for wave resistance is given by

1 V^ R=1-p7rV-/ (P2 + P2 + P2 + P-) cos3 9 d 9 4 Q 1 2 3 4

[28]

Hence in our case C = C + L and s ^b


-19-

TT/S -2k f secre 4 2. R = 16 p ir / [k 4 ^- e

0 0

-k d secrö 5Q , „2/-, _ o ^2 sec 9 + m (1-e

i )2 cos(

-k f sec'ö -k d sec2e ■2k 2 m |j. e (1-e )sec2 0 slnfk a sec9)]d9

o i o

b is Ü1 [29:

where the resistance due to the bulb is

7r/2 -k f sec29 16 p ir k 4 a2 f e sec5

0 9 d9

-k f , \

= ^ TT p e 0 k 4 n2<k (k f) + 1 + ^7 K (k f)) r 000 2k f / 1 0 ' L ' o

from Equations [14] - [17] where K , K , modified Bessel func- 0 1

tlons of the second kind. The resistance due to the line source

is

Tr/2 -k d sec2 9

R = 16 p TT m2 / (1 - e 0 )2 cos 9 de ÜS ! 0

= 16 p TT m2

■k0d/2

1 - k d e 0 \\^o^2) - Ko(kod/2*

-k d + k e 0 ^K (k d) - K (k d))

0 I 1 0 ov 0 '

and the resistance due to the interaction between the bulb and the

line source is

HYDRONAUTICSj Incorporated -20-

Rft 2„ ,, r^/s ,, o 32 p7rk2m \x J

0

-k f sec^e ( -k d sec2^ 1-e 0

X sec2 0 sin (k a sec 0) d0 [30]

which can be evaluated by a series expansion as in Section 6.

11. NONDIMENSIONAL FORM

If we put

_ Vd mi = ra 1 IT ^

Vb-' [31:

where m is a nondiraenslonal coefficient^ we nondimensionalize

with respect to d, e.g.

C = R = R

fpV^d , P, = ~ , F = k d, f

a,a d ^fed o o f d '

- b _ a b = T ^ a ^ T d d

[32]

and dropping bars for convenience we obtain

R = — e'kof JK (k f) + fl + -i . b „a lovo/ 1 2k_f

pdB o / K, (kof)> [33]

Rfl = 2 m' is

l-ic e-^/2 <K o 1 i \ £

K k _c 2

+ k e o

X <K, (k ) - K (k )) 1 i v o' o v oY [34]

Ril = - 8m ^- / P* 0 d

_/ -k f aec20 Va _ o -k sec 0

1-e o sec2 0

x sin (k a sec 0) d0 v o ' [35]


■21-

If we use the notation In Section 5

R.. = - bm .Ri k f

ako, -f- , 2 Ri k (f+l)

aV "^2 ' ? .36]

R = R1 +R/i +R.. b ^s 11 [29]

R.. is an oscillating function of k a., and if we apply the Jyl 0

method of stationary phase (see e.g. Lambj 19^5^ p. 395) we

know that it behaves like

-k f I -knl / TT! e 0 1-e 0 sin k a + f

when k a is large. o

When k a is sufficiently small, R is always negative 0 J>1

because of the property of the function Rl In Section 5*

[38]

12. OPTIMUM PARAMETERS

In this first mode of the function R.. the optimum value

of a is obtained by finding the stationary value of R.. by the XJJL

graphical method as in Case I. The optimum value of b is ob-

tained easily as in Case I, since R is a polynomial with respect

to b.

2F>/

.3 ^

:■„ ,V2 "kof sec v I. "0 k^d sec'

1-e sec 8sin k a sec0 d0 0

^^ Ko(kof) + (1 + 2rf)Kx (V) [39]

Then the total effect of the bulb of this size on the wave re-

sistance becomes


-22-

R, + R.. b ^i

=-8m£

■f7i/2 e-V sec^öL -k0d sec 2Q1 ^

sec20 sinfk a secö) d9 v o

-k0f K (k f) + 1 + ~-r K, (k f) ov o ' 2k f i v o '

[40]

The graphs of the optimum value of a and the optimum value of

b/ n / ra for the obtained optimum value of a are plotted with

respect to P2 for several values of f, the depth of the doublet.,

In Figures 6 and 7. R/m2, (R + R. )/m2J and R /m2 are plotted

versus F2, taking the optimum values of a and b for each P,2 In

Figure 5.

13. SHIP SHAPES

From the boundary condition [6], we can see that the free

surface behaves like a solid wall when the velocity V is small

o < .09 in Equation [6]). Inul (1957) found by

comparing his experimental results with theoretical analyses

that the effect of assuming that the free surface acts like a

solid wall, when computing the stream surface generated by a

given singularity distribution, led to negligible error up to a

Froude number of P = 0.7. Accordingly we have for the poten- Li

tial due to the source line and its mirror image:

d \ df

S -d Vx2+y2+(z-f)

jz+d + Vx2+y2+(z+d)1

=■ = m log, 2 1 U-d +Vx2+y2+(z-d)2/


■23-

Por the doublet

/

- ^(x+a) d

1 3/2 f 1 3/ 2

\

(X+a)2+y2+(Z-ff)^ i(x+a)

2+y2+(z-f) 2V

Considering a sink line from point (LjO,0) to point (L^Oj-d)

In addition to the doublet-source-line system we find for the

total velocity components,

U = -r— +¥,¥ = - T— , W = - ■C— . öx dy dz

Then the streamline equations are

dx dy dz u v w '

We solve this by Runge-Kutta-Gill's (Runge, 1Q95; Kutta, 1901;

Gill, 1951; or see Ralston and Wilf, i960) numerical method to

obtain the three dimensional streamlines starting from the stag-

nation points. Figures 9-11 show the body streamlines repre-

sented by three special cases of different singularity distribu-

tions respectively: vie., a point doublet, a source line, and a

sink line with the three different sets of parameters. Figure

17 shows the case of the doublet line discussed in Case III,

Each of these figures shows the projections of the several body

streamlines on the x y plane and the x z plane„ The calcula-

tions were performed on the IBM 1620. In making these computa-

tions it was necessary to exercise extreme care in the selection

of interval Increments at the neighborhood of the stagnation

point and in the vicinity of large curvature. The resulting

body streamlines are smooth curves with a hollow place although

the distance between the doublet and the source appears to be


.21^-

too large to be a single connected body.

14, DISCUSSION

The optimum distance between the doublet and the source

line., and the optimum size of the bulb Increase with the flow

speed and depth of the doublet as shown In Figures 6 and "J.

However, the depth effect of the doublet on the wave resistance

decreases with flow speed as long as we take the optimum bulb

size and the optimum distance between the doublet and the source

line at each selected depth. This is shown in Figure 8.

The size of bulb is considerably large for the reasonably

high Froude number. When F2 > 2 the width of the bulb becomes

larger than the beam length as shown in Figures 7 and 10,

Hence the bulb may be considered rather as a blunt bow of ship

than an appendage. However, as we notice in Figures 10 - 11,,

the hollow place between the bulb and ship body may contribute

to the flow separation.

Assuming no serious separation takes place the reduction

of the wave resistance due to the bulb is remarkably great.

Figure 5 shows that the effect of an optimum bulb Is to reduce

the wave resistance by more than 60 percent of that contributed

by the line source alone at all Froude numbers.

The wave resistance coefficient of the source line bow

alone is also shown in Figure 5. It has a jump at Fcl= 0, and

it decreases as F,increases. Since this is a nondlmensional

quantity it does not mean that the wave resistance itself has

a jump at F = 0.


-25-

CASE III

15. WAVE RESISTANCE OF THE SYSTEM OF A DOUBLET LINE AND A SOURCE

LINE

As shown In Case II the optimum size of the bulb Is re-

markably large, especially for higher Froude numbers. We will

now Investigate the result when the doublet strength is dis-

tributed linearly along the vertical line from (-a^O,-f, )J) to

(-a^Oj-fg) In front of the same source line as In Case II.

If we take [i = ^ + fu for the doublet strength per unit

length, we have for the wave height far behind the system due

to the bulb, from Equation [26]

8k

V o_ rVs r'a '"o

f_ -k.f sec 9 e 0 (m+M) sec4 0

X [sin (k x secö) cos (k a secö) + cos (k x sec(

x sin (k a secö)] cos (k y sin 0 sec20) d f d

8 TT/ v i k sec20<(e 0 1

o

k f sec 0 (^2f1+|i1)- e

-k0f2sec'

x (u f +M-JK H2 U -kAf sec

20 k f^sec2( O 1 0 2

x [sin (k x sec0) cos(k a sec0) + cos (k x sec0) v o ' v o ' v o '

X sin (k a sec0)] cos (k y sin 0 sec20) d0 [hi]

and due to the source line, from Equation [25]


■26-

8m, 3s V

v2 I -knd sec'0 e I sec 0 cos [k x sec 0J

X cos (k^y sin 0 sec2 0) d0 [25]

Using Havelock's formula (27) and (28), we have for the total

wave resistance, due to the doublet and source lines

R = 16 Trp /^^rV -k d secê]

1 - e COS 0

■2knf. sec 0 + k ^ sec 0<e 0 1

0 \\x fi+M-ir+ e 2 -Ek0f2sec

(H f +u ) 00 J- 2 2

2 ^-k^f.+f^sec2© (^f1+M-1)(M.2f2+M.1)^ 2kc

2knf1sec£0/ x -2k f sec'e xle 0l (n2f1+n1)n2+ e (n2f2+^M2

.2, -kn(f1+f,)secii0 ,

e 0 \2W^S ^+h)\ cos 0

-ak^f. sec'e -2knf;Jsec'10 + a 2 e 0 cos30+n 2 e 0 cos30-2n 2

222

x e -ko(f1-i-f2)sec20

cos30+ ■kr.f-.sec'ê

- e -k0f2sec 0

(f2^+^)j>+2kom1 ■ko(f1+d)sec20

(fâ+M-i )

-kn(f +d) e (f2M.2+M.1)|>-2m1ii2Cos20

' -^^sec2© -kof2sec20

-k (f1+d)sec2e '-ko(f2+d)8ec20 -i e 0 + e sin (koa sec 0£>d0[42]

HYDRONAUTICS^ Incorporated

16. NONDIMENSIONAL FORM

-27-

We perform the Integrations using the methods taken in

Equations [llL [15], and [16]^ and nondlmensionallze physical

variables with respect to m and d as in Case II, i.e.

C = R = R \\ - _ ^2 ._ mVd w M- K TT 2ir2H2 J Ki ~ m d ' ^2 - m' mi "" 4 '

— p m^V d ii 1

2

P^ = -—== , k = k d, f = f./d , f = f /d d "yga o o i 1/ ' s s'

and dropping bars for convenience we obtain

Rib +

'K] -k R„ = 2 is

2

R — R/i + Rn-L. + R n a • is ib iii Ik

1-k« ~ o i \ 2 / „Kl^-V^ê-^^)-^^)

O 2 J ■krtf 0 l -k0f2

-- - K (k f )(\if +[i1 f+ — 2 ov o i 2 i î ' 2

[^3]

44'

R^= P *

XKo(kof2)(.2fA)2 .e-o(fl-2)AKN(fi-2)Ay (^f^^)^^)

+ .4 -i^f k f e 0 !

o 1 K (k f )-K (k f1 ) (^f,-»-^ )[x

1 O 1 O O 1 1 1 1 ' £

■k f. + kf e 0K(kf)-K(kf) (M- f + M- ) M-

02 iôa7 0V02' V 2 2 ^1 ' 's

Kî^J -^ (f +f )/21 / \ i u - 0 0 2 e 0 1 2y/ ^K k (f +f )/2 -K k (f+f )/2

2 ) -1- t 0 1 2 y/ 0 0 v 1 2 ^ | 11 'J X 2nin2+ix2

2(fi+f2)

k„f. r

u ^ 2

+ ^ M-2

J 2

O 1 e 0 1<2k f K (k f )-(2k f -l)^ (k f )lk f I 010v0l/v 0 1 'iôi'ioi

+ e o i

x K

k f <2k f K (k f )-(2knf -l)K (k f ) 0 2 | 020V02'V 02 ' 1K 0 z'

kjr +f )/2 r 0 * 2 ^ k (f +f ) ^k (f.+fj

O V 1 2 ' | 0V 1 2 '

k (f +f )/2 - (k (f+f )- 1) K k (f +f )/2 0V1 2'' V0V1 2 / 10V1 2''

.45]

EmRONAUnCS, Incorporated

■28-

iil P. (*> A ) Rl

k f Rl ak ,

o

k (fn-M)

(^M-^) Rl k f

1 0 2 A ako^ -r > 0 - Rl ko(f2+l)

aV 2 '0

4 n2 ^ Rl k f

o 1

Rl ak , o

ako, 2 ,

k0(f1+l) . - 2

Rl

+ Rl

k f ako, -Y.-2

k (f2+l) ako, -^ . - 2| [46]

where Rl [gjk^n] =? J77"/2 e secn9 sir (g sec 9) do 0

which Is defined In Expression [12],, and Is evaluated In Equa-

tions [12] - [18] by the series expansion In g.

17. OPTIMUM PARAMETERS

The optimum values of u, and LL which make R., + R,.. 1 ^2 ih ill

minimum may be obtained by the usual methods. We differentiate

R partially with respect to |JL and a respectively^ put the re-

sults equal to zeroj and solve the resulting two simultaneous

equations for the optimum values of |i and |x . The equations

for \x and p. resulting from the above mentioned differentia-

tion are

M. = 1

Y - [X X 1 2

W,

V M M- = 2 W. [47]


-29-

where

X = -

d

-k f -k0f 01K(kf)f +e 02K(kf)f

0 O 1 !■ 0V02 2

-M-Vf2)/

2 v e K (k (f +f )/2 (f.+f^) O 1 2 ' ! 2

d L

-k^f -k„f e 0 1 K, (k f ) fn+ e 0 2 K (k f ) f

1V01'1 1^02'^

■ko(f1+f2)/2

K, k (f +f )/2 (f +f ) O 1 2 '' I v 1 2 '

r k f v. - -^- <Rl ak , — , 0

o' 2

k (f +1) . r;iiako, -

2-| , 0

d

-Rl L

k f i av— > 0

+ Ri akoJ ^— 3 0

F. f^Rl

k f Ri

k (f.+l) ako^ 2 . 0

f <Ri i 0 2

ak , o' 2

W, =

- Ri

1

k f

, o| - Ri -1

-Ri ak , 0

k If +1) cr 2 '

ak , , 0 o' 2

k (f+1) o v i '

+ 2<Ri k f

-2 + Ri Vf2+1')

ako, —-2

-kOfl / X -kOf2 / X -ko(fl+f2)/

2

e 01K(kf+e 02K(kf-2e ' ': 0' 0 2

X K k (f^+f )

0 v J- 2 /

ol 2


■30-

W„ = 1 I o 2 0 i

3F. f " e

-knfp K(kf)+f2e K(kf)

ov o 1 p 0X 0 2

2(f12-f1f2+f2

2) -k0(f1+fa)/

3Pd4

2 I K k (f+f )/2

+ 3F

i2 , J -kofx + f MkofJ + r7 + fs

d Lv^d / /

'(fl+f2) I -k^(f +f )/2 — - (f +f ) e 0^ 1 2)/

X K, ko(fl+f2)

2F.

2 /

The optimum value of the distance, a which makes R minimum

Is obtained numerically as In Cases I and II, The optimum values

of a and Li were calculated for various values of f and f . 12 1 2

The results of these calculations are shown In Figures 12 and

13. The wave resistance coefficient for the obtained optimum

parameters for each Proude number Is also calculated, but this

Is the same as that of Case II.

18. EFFECT OF STERN

The stern can be represented as a uniform sink line from

(LJOJO) to (LjO,-l) whose total strength Is the same as the

source line. Here L Is nondlmenslonallzed with respect to d.

The wave resistance due to the source and sink lines alone ex-

cluding Interferences Is as twice as that due to the source line

alone. However, the effect of the Interference between the


-31-

singularities representing the forward and aft part of the ship

may be extremely important. The wave resistance between the

source and sink lines R .. is evaluated using the method of sta- ll

tionary phase (see e.g. Stoker, 1957) in nondimensional form.

R 11 ^ /

^ -k sec'

cos 9 cos (k L sec 0) do

27rl2

k L 1 o /

1 - cos (k L + ~)

which is approximately valid when F, = ,,, JL \/ K JJ

< A (Inui, 1955).

[48]

Similarly the wave resistance due to the interference

between the doublet line and the sink line is

R21 = 2|1- e -k.

1 'e'^1 (^f.+nj -e"^2 (^f.+M-j; P 2 d

-k fi -knf i ' + H2 e 0 1 -e 0 2

FISLT sin^0(

a+L) + f; [49]

The optimum values of [i. and \i change very slightly when the

sink line stern is considered (see Table l).

The total resistance for the system including the sink

line stern and optimum doublet lines is plotted in Figure 15 for

Froude numbersj P = 0.2, 0,25, and 0.3- The numerical values

of the stern effect for each Froude number are also shown in

Table 1. Here we can see that the interference between the

stern and the bow plus bulb is in general much less than that

of the stern and the bow alone. That is, the bulb has an effect


-32-

of smoothing out the humps and hollows of the resistance curve

to a considerable extent. Besidesj the magnitude of the inter-

ference resistance between the bulb and the bow is much larger

than either the interference resistance between the stern and

the bow or between the stern and the bulb.

19. DISCUSSION

Under the optimum conditions, the effect of the line

doublet is similar to that of the point doublet of equal total

strength. The optimum position of the center of gravity of the

line doublet increases in depth and mo es forward with increas-

ing Froude number while at the same time increasing in total

strength. This is shown by Table 1, Figures 12 and 13. The

total effect of the line doublet on the wave resistance is al-

most exactly the same as that of the point doublet.

The wave resistance curves due to the systems of source

line and the doublet line alone optimized for Froude numbers

PT = 0.2, 0.25J and 0.3 are shown in Figure 14. The wave re-

sistance at a little lower Froude numbers than the optimum

Froude numbers is larger than that due to the source line alonej

but at Froude numbers in the vicinity of and greater than that

for which the line doublet was optimized, the effect of the line

doublet (bulb) is always to reduce the wave resistance.

The variation of the strength of the doublet line (assumed

to be linear) is very small In general unless we make the

doublet line long enough so that its upper end is very near to

the free surface. In this case the optimum slope Is quite large

for low Froude numbers. However, the upper end of the linear

doublet line should not be too close to the free surface for op-

timum interference. In fact, It can be seen from Table 1 that


-33-

the optimum depth of the upper end Increases with Increasing

Proude number.

20. APPLICATION

If we want to Improve the ship shape of length hs whose

waterllne is given by the polynomial (G. P. Weinblum., 1950) as

A n n+S n+1

n=0

then the singularity distribution in the sense of Michell' s

ship is

5 n

A X n

n-0

The height of bow wave at a large x is given by (see

Equations [25], [26] ),

kl a. -k^z1seG 3bs Vk

o 0

2! a +

k ssecii0 k ^sec' 0 0

x sin (k x sec 9) + 3i a, 51 a,

+ k secö , ^ 3. , .; 5^ 0 k sec^Q k sec 0

o o

x cos (k y sin 0 sec2 0) d0

cos(k x sec0))>

[50]

If we combine the bulb whose wave height at a large x is

8k 2

C n ^ u/ -kj sec^O ,

- —^— r'/2 sec40 e 0 <^sin (k x sec0) cos (k a sec0) Y J 1 v 0 ' x 0

+ cos (k x sec0) sin (k a sec0) cos (k y sinQ seG2©)^ d0

the wave resistance due to this bulb and bow are obtained from


Equation [29] and are given by

R 0 \L'

■k z sec' o 0 4] a.

+ ko k 3sec2e k 5sec4(

o o

Hk 2 e " 0

o

k^f sec2f sec49 cos (k a sec 0) v o '

cos3 6

+ cos" -k z sec

e i

31 a. 5! a. +

-k f sec 9 t l2\

uk 2 e 0 sec40 sin (k a sec e)i J> d0 ^ o v o ' ^

All the Integrals invol/ed here are of the form

k 2sec0 k 3sec30 k 5sec50 ' ooo

TT/ -2h sec 0 an+i I = / /2 e sec 0d0 (shown In Equation [Ik]) n 0

If we consider the expansion of cos (k a sec9) and sin (k a sec0).

Hence the wave resistance in this case may be evaluated exactly

in the same manner as in Section 5.

The optimum parameters, f, \x, and a may be determined by

the methods used in Sections 7 and 12.

If we Include the stern the expression for the wave re-

sistance becomes a little complicated. However., the forms of

the Integrals are the same as those encountered previously when

the method of stationary phase was used for F < 0.4.

Since the effect of the bulb is largely due to the inter-

ference of the bulb with the bow wave (Takahel, I960), we need

not be concerned about the Influence of the stern in dealing

with the determination of the optimum bulb parameters even though

they vary slightly as shown In Section 18.


-35-

The elementary bow waves (Havelock, IQS^a) of usual ships

are combinations of sine waves and cosine waves as shown in

Equation [50]. Since the best position of the doublet for the

sine bow waves can be easily shown to be at the bow itself, the

distance between the doublet and the bow in the case of usual

ships will not be as large as is in the case of a source line

ship. Hence the series expansion used in finding the distance

may be evaluated more easily.

HYDRONAÜTICS, Incorporated

-36-

APPENDIX

Consider

, rTr/2 -2h sec 0 2V

Ri (t.h^v) = jl/z e 0

sec2 0 sin (t sec 0) d0,

Expanding sin (t sec 0) we have for the integrand

V (-iMt sec 9)2n+X -2h sec2e l^ (2 n+l) 1 e

n=0

2 V sec 0 + R ,

m

2m+3 , 2^ , , (t sec0 -2h sec^e 2V

where R is the remainder and R < /g ^ \—j— e sec 0 [Al]

In 0 <_ 0 < 7r/2 by Taylor's remainder theorem. It then follows

that

m r ^ -h v ' t

2n+i ! / Ri (t.h^v) = J K (hj-K1 (h)> + r2 R d0 .

2 -n (2n+l)!2n+V 0 ' o{ \ .6

n=0 (n+v)

To prove

Ri (t.h^v) =

n=0

sn+i ^K (h)-K,(h) s

(n+v)

we have only to prove

llm f7^2 R d0 = 0 . m -► oo J m

By the inequality [Al]

j^2 R d0 J m

^ (7r/2 -2h sec20 t2m+3sec2v+2m+30 ^ < J e ——-r\—r— - d0

0 (2m+3)

t2m+3{K (h)-K'(h) , o ov ^(m+x + v)

(2m+3); 2 m+i + v


-37-

Nextj for later use we have to prove an Inequality

v-i K (h) <

h VI K^h)

where 0 < h < 1 and v is a positive Integer.

From the recurrence formulae

K (h) = f^ K (h) + K (h) v+iv ' h vv ' v-iv '

Suppose we have the inequality [A2] for v and v-i. Putting

inequality [A2] into Equation [A3], we get

V V-2

K (h) < — wl K, (h) + (v-i)! K^h) v+i v ' — v i ^ ' -y-a x ' i x ' h h

^iil , K (h)

But we know

[A2]

:A3]

K2(h) < 2-2 ! Kjh)

Hence by the mathematical induction, the inequality [A2] is true,

From the recurrence formula

- K ' (h) = i K (h) + K Ln (h) V v ' 2 V-l ' V+l

we can prove

(-1) v d K (h)

dh < K (h) + K (h) + ... + K (h) when v is — V V-2 v ' 1 x '

odd

[A4]

< K (h) + K (h) + ... + K (h) when v is even|


-38-

By definition

K (h)-K'(h). . ov ' ov '(n) \^)-0

dK (h) d2K (h)

1' dh + y dh'

+ (-1) n dnK (h)

dh n [A5]

Using the inequalities [A2] and [k^] in Equation [A5] we can prove

n-i

^o(h) KH ^ I—— MM for v < h< 1

(n)

J^2 R d0 m <

jm+3 m«miK1(h)

hm 22+v(2ra+3)]

The left hand side will approach zero when m becomes larger than

t2/ho When h > 1 the proof is simpler.

HYDRONAÜTICS, Incorporated

-39-

REFERENCES

1. Eggart,, E. ¥., "Form Resistance Experiment''^ SNAME^ 1935»

2. Gill, S. A., "A Process for the Step-by-Step Integration of Differential Equations in an Automatic Digital Computing Machine''^ Proc. Cambridge Philos. Soc.,, Vol. 47, pp. 96- 108, 1951.

3. Havelock, T. H., "The Wave Pattern of a Doublet in a Stream", Proc. Roy. Soc, A Vol. 121, p. 515-23, 1928j "Wave Patterns and Wave Resistance", T.I„N.A., Vol. 76, pp. 43G-443, 1934a; "The Calculation of Wave Resistance", Proc. Roy, Soc, A Vol. 144, pp. 514-521, 1934b.

4. Inui, T., "Asymptotic Expansion Applied to Problems in Ship Waves and Wave-Making Resistance", Proc, of the 5th Japan National Congress for App. Mech., 1955.

5. Inui, T., 60th Anniversary Series, Vol. 2, i960.

6. Inui, T., Takahel, T. and Kumano, M., "Wave Profile Measure- ments on the Wave-Making Characteristics of the Bulbous Bow", Society of Naval Architecture of Japan, i960.

7. Kutta, W,, "Beitrag zur näherungsweisen Integration totaler Differential glelchungen", Z. Math. Phys,, Vol. 46, pp. 435- 453. 1901.

8. Lagally, M., "Berechnung der Kräfte und Momente, die strömende Flüssigkeiten auf ihre Begrenzung ausüben", Zeitschrift Für Angewandte Mathematik und Mechanik, Band 2, 1922.

9. Lamb, H., "Hydrodynamics", Dover Pub., New York, 1945.

10. Lunde, J,, K., "On the Theory of Wave Resistance and Wave Profile", Skipsmodelltankens meddelelse Nr. 10, 1952.

11. Milne-Thomson, L. M., "Theoretical Hydrodynamics", 4th ed., Macmillan Company, New York, 1956,

12. Ralston, Anthony and Wllf, Herbert S., "Mathematical Methods for Digital Computers", John Wiley and Sons, Inc., New York, I960.

13- Runge, C, "Über die numerische Auflösung von Differential- gleichungen, Math. Ann, Vol. 46, pp. 167-I78, 1895.

14. Saunders, H. E., "Hydrodynamics in Ship Design", Vol. 1, pp. 368-371, 1957.


40-

15. Stoker^ J. J., "Water Waves", Intersclence Publishers, Inc., New York, 1957-

16. Takahel, T., "A Study on the Waveless Bow", SNAJ, i960.

17. Taylor, D. W., "Some Model Basin Investigations of the In- fluence of Form of Ships on Their Resistances", SNAME, 1911.

18. Taylor, D. W., "The Speed and Power of Ships";, 3rd ed. U. S. Gov't Printing Office, Washington, D. C, 19^3.

19. Weinblum, G. P., "Die Theorie der Wulstschiffe", Der Gesells- schaft für Angewandte Mathematik, 1935-

20. Welnblura, G. P., "Analysis of Wave Resistance", DTMB, Report 710, 1950.

21. Wigley, W.C.S., "The Theory of the Bulbous Bow and its Practical Application", Trans. N.E.C.I.E.Sc, Vol. LII, pp. 65-88, 1936.

HYDRONAÜTIGS, Incorporated

TABLE la,

OPTIMUM PARAMETERS AND THE WAVE RESISTANCE (d/L = O.O?)

PL • 15 .2 .25 • 3

^xo .1^^47 7.5706-02 9.2753-02 .24825

^0 -1.2661-04 -3.O208-O3 7.2169-04 -2.9446-05

fx .02 .02 .02 .04

f2

• 03 • 07 .07 .07

a • 0295 .048 .06875 .09792

R«b 24.108 20.480 15.756 II.747

Rls 30.818 26.682 21.389 16.697

RM1 -48.217 -40.960 -31.513 -23.493

R 6.7090 6.2019 5.6327 4.9505

Rxi -3.4890 -8,7006 4.3106 -5.5123

R21 5.9074 10.170 -6.3734 2.5322

Rt 39.9452 34.353 24.959 18.667

^xs .12678 5.7288-02 .11140 .22150

^as ~3.0604 -I.2358-02 3.6163-03 -1.3006-04

LL , LL : Values of a, and LL In the negative optimum doublet 10 20 ^ 2

strength \i = \i + f\x } without considering the stern.

7.5706-02 = 7.5706-10~2

R„^R,, JR«^! Wave resistance due to the doublet line, the ib is' Hi source line., and the interference defined by Equations [45L [^4], and [46] respectively

R = Ris + Rib + RUx

R, .jR .: Wave resistance interference between the source and Xl 21

sink lines, and between the doublet and sink lines respectively

R, : Total wave resistance including the stern with u. and LL t D ^io 20

a jU, : Optimum values of u, and u^ when the stern is considered riS 2S X ^2


TABLE lb.

OPTIMUM PARAMETERS ! AND THE WAVE RESISTANCE (d/L = O.07)

FL • 35 .4 .45 .5

^xo .28731 .32629 .36552 1.4464

^0 -I.7513-O5 -2,1465-05 -I.7576-05 2,8336-07

f .04 .04 .04 ,06

f2

• 07 .07 .07 • 07

a .1145 .127 .1405 .15

Rib 8.4268 6.5796 5.0410 3.9724

Ris 13.043 10.311 8.2791 6.7546

Riil -16.853 -13.159 -IO.082 -7.9447

R 4,2683 3.7315 3.2381 2.7823

Rxl 4,7276 -2.9417 -2.6130 -.17448

Ral -1.8833 2.9071 .26984 -1,0252

Rt 20.160 14,008 9.1740 Ö.3372

^xs .31814 ,25420 .35574 1.6330

^s 3.8745-05 -9.3216-05 -2.3353-05 7.9889-07

LL ^u : Values of u, and u. In the negative optimum doublet

strength \i = \i + f\i , without considering the stern.

7.5706-02 = 7.5706.10"2

R .,,R.. jR... : Wave resistance due to the doublet line, the

source line, and the Interference defined by Equations [45], [44], and [46] respectively

R - Ris + R^b + Rm

R.,.^ .: Wave resistance interference between the source and xi si sink lines, and between the doublet and sink lines respectively

R, : Total wave resistance including the stern with 11 and ix t 10 20

XS ^2S Optimum values of \x and |x_ when the stern is considered

HYDRONAUTICS, INCORPORATED

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HYDRONAUTICS,INCORPORATED

> Q.

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FIGURE 5-WAVE RESISTANCE OF A SOURCE LINE AND OPTIMUM DOUBLET LINE AT EACH FROUDE NUMBER

6.0

4.0

a

2.0

1,0 2.0 3.0

d ~gd


4.0

1

i-

^ ^

1.0 1

0.6 _—

0.2

A ^> "

5,0

FIGURE 6- OPTIMUM DISTANCE BETWEEN DOUBLET AND SOURCE LINE

A Point Doublet Atx=-a,y:0, z=-f

A Source Line At x-0, y:0, -d<z <0

6.0

0.5

^rr 3—-n

^H 0.4

-—- ____2i——J

1.0 2.0 3.0 4.0 3.0

d gd

6.0

FIGURE 7-OPTIMUM RADIUS OF BULB (A POINT DOUBLET)

A Point Doublet At x =0, y:0, z =-f

A Source Line At x = 0, y = 0, -d < z < 0


FIGURE 8- OPTIMUM DEPTH OF DOUBLET


FIGURE 9- BODY STREAM-LINE SHAPE DUE TO A SOURCE LINE AND A POINT DOUBLET


-020 -0.15 -0.10 -005 0 0.05

FIGURE 10- BODY STREAM-LINE SHAPE DUE TO A SOURCE LINE AND A POINT DOUBLET


0.08

Y L t

[^ L- L—

0.04

0

0.04

0.08

-m?

/ 0 *J,* ► x

1 POI

® ^JT DOUBLE"

^ SOURCE'LINE ^L

\

^

-0 20 -0. 6 -0. 2 -0. 3 -0 4 4 0.4 0.8 0.12 0.16 0.20

-0.04

-0.08

—Hi

M/l/V

1 L

FIGURE IIA

SINK LINE

FIGURE II- BODY STREAM-LINE SHAPE OF SOURCE AND SINK LINES AND A POINT DOUBLET

d =0.035 //' = 1.42772x10

m'=0.0175 FL=0.37

a =0.10395 f =0.03

HYDRONAUTiCS, INCORPORATED

0.20

T- 0,10

f, = 0.04, f^ÔO^-—■ -^ r^

Uss hro-os/ f2=0.07

fr0.02, | f2= 0.05

0.15 0.20 0.25 0.30 0.35

F-^k l^gT

0.40 0.45 0.50

FIGURE 12-OPTIMUM DISTANCE a, BETWEEN DOUBLET LINE AND SOURCE LINE

Doublet Line At x=a , y=0,-f2<z <-f|,-y-'0.07, L= 1.0

0.40

0.20

fÔ.02, f2=0£3^-^

fÔ.04, f2=ao7--—-

^—

fÔ.02, f2= 0.05

= f^Ü.02, f2=0.07

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

F = j^T

FIGURE 13-OPTIMUM STRENGTH OF DOUBLET LINE ß-U^-ißz

Doublet Line At x = o, y»0,-f2<z«-f| ^ =0.07^=1.0

//- Is Extremly Small As Shown In Table I, x = a Is Shown In Figure 12


0.15 0.20 0.25 0.30 0.35 0.40

KgL

0.45 0.50

FIGURE 14-WAVE RESISTANCE OF SOURCE LINE AND DOUBLET LINE (BULB)

Doublet Line At x = -0,y = O,-f2<z -f. With Strength n-u^ -uz z, — = 0.07, L^I.O

F =0.20 (Optimum) F =0.25 F =0.30

A, = 0.0757 0, =0.0927 ^ =0.2482

^2:-3.026 -lO"3 ^2= 7.2169-lO"4 ^2 =-2.9446 • 10" 5

f, =0.02, f2:0.07 f, = 0.02, f2= 0.07 f^O.04, f2=0.07

a =0.048 a = 0.06875 o =0,0979


120

E

b|<v

100

80

60

40

Z1 ^

~ -- 0.07

d - LENGTH OF SOURCE OR SINK LINE

L = DISTANCE BETWEEN SOURCE LINE AND SINK LINE

R = WAVE RESISTANCE

m = STRENGTH OF SOURCE PER UNIT LENGTH

WITH OPTIMUM BULB AT F = 0,2

20

OPTIMUM AT F= 0.25

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0,50

F = yOO

FIGURE 15-WAVE RESISTANCE FOR SOURCE LINE SHIP INCLUDING STERN

(Sink Line)


ID

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Vgl

FIGURE 16-WAVE RESISTANCE INTERFERENCE WITH STERN (Sink Line)


FIGURE 17-BODY STREAM-LINE SHAPE DUE TO A DOUBLET LINE AND A SOURCE LINE

Strength Of Doublet /^/^-/y, At X = Q, y =0,-f2<z, —f, ,L=I.O,-^=0.07


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